Previous |  Up |  Next

Article

Keywords:
Stability; Lyapunov functional; delay differential equations; third-order differential equations
Summary:
In this article, we shall establish sufficient conditions for the asymptotic stability and boundedness of solutions of a certain third order nonlinear non-autonomous delay differential equation, by using a Lyapunov function as basic tool. In doing so we extend some existing results. Examples are given to illustrate our results.
References:
[1] Burton, T. A.: Stability and periodic solutions of ordinary and functional differential equations. Mathematics in Science and Engineering 178, Academic Press, Orlando, FL, 1985. MR 0837654 | Zbl 0635.34001
[2] Burton, T. A., Makay, G.: Asymptotic stability for functional differential equations. Acta Math. Hung. 65, 3 (1994), 243–251. DOI 10.1007/BF01875152 | MR 1281434 | Zbl 0805.34068
[3] Greaf, J. R., Remili, M.: Some properties of monotonic solutions of $x^{\prime \prime \prime }+p(t)x^{\prime }+q(t)f(x)=0$. Pan. American Math. J. 22, 2 (2012), 31–39. MR 2934271
[4] Omeike, M. O.: New results on the stability of solution of some non-autonomous delay differential equations of the third order. Differential Equations and Control Processes 2010, 1 (2010), 18–29. MR 2766411
[5] Omeike, M. O.: Stability and boundedness of solutions of some non-autonomous delay differential equation of the third order. An. Stiint. Univ. Al. I. Cuza Iasi. Mat. (N.S.) 55, 1 (2009), 49–58. MR 2510712 | Zbl 1199.34390
[6] Sadek, A. I.: On the Stability of Solutions of Some Non-Autonomous Delay Differential Equations of the Third Order. Asymptot. Anal. 43, 1-2 (2005), 1–7. MR 2148124 | Zbl 1075.34075
[7] Sadek, A. I.: Stability and Boundedness of a Kind of Third-Order Delay Differential System. Applied Mathematics Letters 16, 5 (2003), 657–662. DOI 10.1016/S0893-9659(03)00063-6 | MR 1986031 | Zbl 1056.34078
[8] Swick, K.: On the boundedness and the stability of solutions of some non- autonomous differential equations of the third order. J. London Math. Soc. 44 (1969), 347–359. DOI 10.1112/jlms/s1-44.1.347 | MR 0236482 | Zbl 0164.39103
[9] Tunç, C.: On asymptotic stability of solutions to third order nonlinear differential equations with retarded argument. Communications in applied analysis 11, 4 (2007), 515–528. MR 2368199 | Zbl 1139.34054
[10] Tunç, C.: On the asymptotic behavior of solutions of certain third-order nonlinear differential equations. J. Appl. Math. Stoch. Anal. 2005, 1 (2005), 29–35. DOI 10.1155/JAMSA.2005.29 | Zbl 1077.34052
[11] Tunç, C.: On the stability and boundedness of solutions to third order nonlinear differential equations with retarded argument. Nonlinear Dynamics 57 (2009), 97–106. DOI 10.1007/s11071-008-9423-6 | MR 2511159 | Zbl 1176.34064
[12] Tunç, C.: Some stability and boundedness conditions for non-autonomous differential equations with deviating arguments. E. J. Qualitative Theory of Diff. Equ. 2010, 1 (2005), 1–12. MR 2577154
[13] Tunç, C.: Stability and boundedness of solutions of nonlinear differential equations of third-order with delay. Differential Equations and Control Processes 2007, 3 (2007), 1–13. MR 2384532
[14] Yoshizawa, T.: Stability theory by Liapunov’s second method. The Mathematical Society of Japan, Tokyo, 1966. MR 0208086
[15] Zhu, Y. F.: On stability, boundedness and existence of periodic solution of a kind of third order nonlinear delay differential system. Ann. Differential Equations 8, 2 (1992), 249–259. MR 1190138 | Zbl 0758.34072
Partner of
EuDML logo